Dirichlet normal-mixture posterior contraction

Put $d_n=\max \{(n{\sigma }_n{)}^{-1\mathrm{/}2}\log n,{\sigma }_n^2\log n\}$. By [F3], choose $C_1\ge 1$ so the prior mass is at least $e^{-C_{1}n{d}_n^2}$ on a set where both local quantities are at most $C_1{d}_n^2$. Set ${\bar{d}}_n=\sqrt{C_1}{d}_n$. Then the mass is at least $e^{-n{\bar{d}}_n^2}$ and both quantities are at most ${\bar{d}}_n^2$. By [F4], except on an event of probability at most $(n{\bar{d}}_n^2{)}^{-1}$, the likelihood denominator is at least $e^{-3n{\bar{d}}_n^2}$. Also $n{\bar{d}}_n^2\ge C_1{\sigma }_n^{-1}(\log n{)}^2\to \mathrm{\infty }$.

Choose a compact interval $I=[L,U]$ containing $\mathrm{supp}(p_0)$ in its interior and contained in the open interval where the density of $\alpha$ is continuous and strictly positive. Such an interval exists because the support is compact; its base density is bounded below on $I$. Put $R=(U-L)\mathrm{/}2>0$ and $\kappa =\max \{a_1,1-4a_2,0\}<1$. The polynomial bandwidth bounds give $n{\bar{d}}_n^2\le C{n}^{\kappa }(\log n{)}^2$. Choose $s_n=c_0(\log n{)}^{-1\mathrm{/}2}$ with $c_0$ large enough that for some $\theta <1-\kappa$, $q_n:=2\mathrm{\Phi }(-R\mathrm{/}{s}_n)\ge n^{-\theta }$ eventually: integrating the standard normal density on $[R\mathrm{/}{s}_n,R\mathrm{/}{s}_n+1]$ gives the claimed lower bound once $R^2\mathrm{/}(2c_0^2)<\theta$. Hence $n{q}_n-5n{\bar{d}}_n^2\to \mathrm{\infty }$. Set $r_n=M_0{ϵ}_n$ for a fixed sufficiently large $M_0$, and ${\xi }_n=\min \{r_n,1\mathrm{/}4\}\mathrm{/}\sqrt{8}$. We have $n{\xi }_n^2\to \mathrm{\infty }$ and ${\xi }_n^{-2}\le Cn$: if $r_n<1\mathrm{/}4$, use ${ϵ}_n\ge n^{-1\mathrm{/}2}\log n$ for all large $n$, and otherwise ${\xi }_n$ is fixed. Choose $d>0$ with $d^2\mathrm{/}(2c_0^2)>4$ and $A\ge 1$ large enough that $I_d=[L-d,U+d]\subseteq [-A,A]$. Then $n{\xi }_n^{-2}{s}_n^{-1}{e}^{-d^2\mathrm{/}(2s_n^2)}\to 0$. Applying [F5] gives posterior concentration on $\{F(I_d^c)\le {\xi }_n^2,\sigma \le s_n\}$. Since $[-A,A{]}^c\subseteq I_d^c$, the larger sieve $S_n=\{F([-A,A{]}^c)\le {\xi }_n^2,\sigma \le s_n\}$ also has ${\mathrm{\Pi }}_n(S_n^c\mid X_{1:n})\to 0$.

Partition $S_n$ into dyadic scale bands $S_{n,j}=S_n\cap \{2^j{\sigma }_n\le \sigma <2^{j+1}{\sigma }_n\}$ for $2^j{\sigma }_n\le s_n$. Their prior masses are bounded by $G([2^j,2^{j+1}))$. Since ${\xi }_n^2=\min \{r_n,1\mathrm{/}4{\}}^2\mathrm{/}8$, [F6] applies at radius $r_n$ and gives $e^{-n{r}_n^2}{\sum }_j\sqrt{N(r_n,\{p_{F,\sigma }:(F,\sigma )\in S_{n,j}\},h){\mathrm{\Pi }}_n(S_{n,j})}⟶0.$ For compactly supported $G$, only finitely many bands have positive prior mass and the first rate term suffices.

If $G$ is compactly supported, put ${ϵ}_n^{\mathrm{c}}=\max \{(n{\sigma }_n{)}^{-1\mathrm{/}2}\log n,{\sigma }_n^2\log n\}$, $r_n^{\mathrm{c}}=M_0{ϵ}_n^{\mathrm{c}}$ and ${\xi }_n^{\mathrm{c}}=\min \{r_n^{\mathrm{c}},1\mathrm{/}4\}\mathrm{/}\sqrt{8}$. Since ${\sigma }_n\to 0$, $n({ϵ}_n^{\mathrm{c}}{)}^2\ge {\sigma }_n^{-1}(\log n{)}^2\to \mathrm{\infty }$ and $({ϵ}_n^{\mathrm{c}}{)}^{-2}\le Cn$. Thus $n({\xi }_n^{\mathrm{c}}{)}^2\to \mathrm{\infty }$ and $({\xi }_n^{\mathrm{c}}{)}^{-2}\le Cn$. Also ${\bar{d}}_n\le \sqrt{C_1}{ϵ}_n^{\mathrm{c}}$. Retain $I,s_n,d,A$ from step 1.2: its denominator, leakage bound, and exponential factor do not depend on the target radius, so the hypotheses of [F5] hold with ${ϵ}_n^{\mathrm{c}}$ and ${\xi }_n^{\mathrm{c}}$ in place of ${ϵ}_n$ and ${\xi }_n$. It yields posterior sieve complement convergence for $S_n^{\mathrm{c}}=\{F([-A,A{]}^c)\le ({\xi }_n^{\mathrm{c}}{)}^2,\sigma \le s_n\}$. The compact branch of [F6] supplies the weighted entropy condition at radius $r_n^{\mathrm{c}}$ using only the first rate term.

The prior is the measurable DP law from [F1], and [F2], [F8] give its jointly measurable density likelihoods. We have ${\bar{d}}_n\le \sqrt{C_1}{ϵ}_n$, so increasing $M_0$ if necessary ensures ${\bar{d}}_n\le r_n$; also $n{\bar{d}}_n^2,n{r}_n^2\to \mathrm{\infty }$. Thus [F3] supplies the local prior-mass condition for [F7]. For general $G$, steps 1.2 and 1.3 provide a sieve with vanishing posterior complement and its weighted entropy condition. If $G$ is compactly supported, step 2.1 instead provides these two conditions at radius $r_n^{\mathrm{c}}$, and its bound ${\bar{d}}_n\le \sqrt{C_1}{ϵ}_n^{\mathrm{c}}$ allows $M_0$ to be chosen so ${\bar{d}}_n\le r_n^{\mathrm{c}}$. Apply [F7] with local radius ${\bar{d}}_n$ and target radius $r_n$ in the general case, or $r_n^{\mathrm{c}}$ in the compact case. It follows that for every sufficiently large fixed $M_1$, the posterior outside $M_1{M}_0{ϵ}_n$ (respectively $M_1{M}_0{ϵ}_n^{\mathrm{c}}$) tends to zero in $P_0^n$-probability. Since $M_1{M}_0$ can be any sufficiently large fixed constant, this proves the stated rates. The measurability and dominated-likelihood requirements are supplied by [F1], [F2], and [F8], and [F9] fixes the conclusion's metric convention.